Question

Two identical metal beads, each of mass M but negligible width, can slide along a thin smooth uniform horizontal rod of mass M and length L. The rod is capable of rotating about a vertical axis passing through its centre. Initially, the beads are almost touching the axis of rotation and the rod is rotating at speed of 14 rad/s. Find the angular speed of the system when the beads have moved up to the ends of the rod. (Assume that no external torque acts on the system.)

Answer 1

Data : ω₁ = 14 rad/s

The MI of the rod about a transverse axis through its CM,

I_rod = \(\frac{ML^2}{12}\)

Since the beads are almost particle-like, and initially touching the rotation axis, their MI about the vertical axis is taken to be zero. When the beads move upto the ends of the rod, r = L/2, their MI about the vertical axis is

By the principle of conversation of angular momentum,

\(I_2\omega_2 = I_1\omega_1\)

The final angular speed of the system,